Minimizing the least eigenvalue of unbalanced signed unicyclic graphs with given girth or pendant vertices

Abstract

A signed graph $\Gamma=(G,\sigma)$ consists of an underlying graph $G=(V,E)$ with a sign function $\sigma:E\rightarrow\{1,-1\}$. Let $A(\Gamma)$ be the adjacency matrix of $\Gamma$. Let $\lambda_1(A(\Gamma))\geq\lambda_2(A(\Gamma))\geq\cdots\geq\lambda_n(A(\Gamma))$ be the spectrum of the signed graph $\Gamma$, where $\lambda_n(A(\Gamma))$ is the least eigenvalue of $\Gamma$. Let $\mathcal{U}^-_{n,g,k}$ denote the set of all the unbalanced signed unicyclic graphs with order $n$, girth $g$ and $k$ pendant vertices, let $\mathcal{U}^-_n(k)$ denote the set of all the unbalanced signed unicyclic graphs with $n$ vertices and $k$ pendant vertices, and let $\mathcal{U}^-_{n,g}$ denote the set of all the unbalanced signed unicyclic graphs with order $n$ and girth $g$. Obviously, $\mathcal{U}^-_n(k)=\bigcup\limits_{g=3}^{n-k}\mathcal{U}^-_{n,g,k}$ and $\mathcal{U}^-_{n,g}=\bigcup\limits_{k=0}^{n-g}\mathcal{U}^-_{n,g,k}$. In this paper, we determine the signed unicyclic graphs whose least eigenvalues are minimal among all the graphs in $\mathcal{U}^-_{n,g,k}$, $\mathcal{U}^-_n(k)$ and $\mathcal{U}^-_{n,g}$, respectively.